Bifurcation And Control Of Biological Systems With Population Effects

In general,there are various interactions between biological populations,and the preypredator relationship is considered very important.In order to better describe the dynamics of intra-species or inter-species relationship in line with the actual situation,factors such as Allee effect,fear effect,time delay,linear harvesting and other factors should be taken into the biological population systems.Therefore,this paper proposes two classes of multi-factor biological population systems and studies their local stability,Hopf bifurcation,and optimal control problems.In Chapter 2,a delayed phytoplankton-zooplankton system with Allee effect and linear harvesting is proposed,where phytoplankton species protects themselves from zooplankton by producing toxin and taking shelter.First,the existence and local stability of the possible equilibria of system are explored.Next,the existence of Hopf bifurcation is investigated when the system has no time delay.What's more,the stability of the limit cycle is demonstrated by calculating the first Lyapunov number.Then,the condition that Hopf bifurcation occurs is obtained by taking the time delay describing the maturation period of zooplankton species as a bifurcation parameter.Furthermore,based on the normal form theory and the central manifold theorem,we derive the direction of Hopf bifurcation and the stability of bifurcating periodic solutions.In addition,by regarding the harvesting effort as control variable and employing the Pontryagin's Maximum Principle,the optimal harvesting strategy of the system is obtained.Finally,in order to verify the validity of the theoretical results,some numerical simulations are carried out.In Chapter 3,a delayed prey-predator-scavenger system with fear effect and linear harvesting is proposed.First,we discuss the existence and local stability of all possible equilibria.Next,we investigate the existence of Hopf bifurcation of the delayed system by regarding the gestation period of the scavenger as a bifurcation parameter.Furthermore,we obtain the direction of Hopf bifurcation and the stability of bifurcating periodic solutions by using the normal form theory and the central manifold theorem.In addition,we give the optimal harvesting strategy of delayed system based on the Pontryagin's Maximum Principle with delay.Finally,some numerical simulations are carried out to verify our theoretical results.